Finite Group Databasebeta

← #127017

Subgroups — #127017

Order 512.

610 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20629C211yesmaximal, characteristic
2#17361C211yesmaximal, characteristic
3#17349C211yesmaximal, characteristic
4#7188C214yesmaximal, complemented
5C2×C4×#41C2211yescharacteristic
6#2920—21no—
7C2×#431—21no—
8C2×#423C411yescharacteristic
9C22×C4⋊C8C411yescharacteristic
10C2×#408C412yes—
11#1798—21no—
12#1798—21no—
13#1797—21no—
14#1797—21no—
15#1729C2212yes—
16#1467—21no—
17#1466C412yes—
18#1465—21no—
19#1465C412yes—
20#1461—21no—
21#1460C412yes—
22#1459—21no—
23#1459C412yes—
24#1458—21no—
25#1458—21no—
26#1458C2212yes—
27#1458C2212yes—
28C22×#41C2×C411yescharacteristic
29C22×M16C2×C412yes—
30C23×C8C2×C412yes—
31C4×#41—41no—
32C4×#41—41no—
33C4×#41—41no—
34C4×#41—41no—
35C2×C4×Q8D411yescharacteristic
36C2×C4×Q8Q811yescharacteristic
37C2×C4×D4C2×C411yescharacteristic
38C2×C4×D4C2×C411yescharacteristic
39C2×C4×D4C2×C411yescharacteristic
40C2×C4×D4C2×C411yescharacteristic
41C2×C4×D4D411yescharacteristic
42C2×C4×D4Q811yescharacteristic
43C2×#117D411yescharacteristic
44C2×#117D411yescharacteristic
45C2×#117C2311yescharacteristic
46C22×C42C2×C411yescharacteristic
47C22×C42D411yescharacteristic
48C22×C42D411yescharacteristic
49#432—22no—
50#432—41no—
51#431—22no—
52#431—22no—
53#431—22no—
54#431—41no—
55#431—41no—
56#431—41no—
57#423—41no—
58#423—41no—
59#423—41no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—41no—
65C2×C4⋊C8—41no—
66C2×C4⋊C8—41no—
67#408—44no—
68#407—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C4×M16—41no—
73C2×C4×C8—41no—
74#336—22no—
75#336—22no—
76#324—22no—
77#324—22no—
78#324—22no—
79#324—22no—
80#323—22no—
81#323—22no—
82#323C2×C414yes—
83#323C2×C414yes—
84C2×#41—22no—
85C2×#41—22no—
86C2×#41—22no—
87C2×#41—22no—
88C2×#41—41no—
89C2×#41—41no—
90C22×Q8C4⋊C411yescharacteristic
91C22×D4C4211yescharacteristic
92C22×D4C4211yescharacteristic
93C22×D4C4⋊C411yescharacteristic
94C23×C4C4211yescharacteristic
95C23×C4C4211yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22⋊C411yescharacteristic
100C23×C4C4⋊C411yescharacteristic
101C23×C4C4⋊C411yescharacteristic
102C23×C4C22×C411yescharacteristic
103C2×M16—24no—
104C2×M16—42no—
105C2×M16—42no—
106C2×M16—42no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—24no—
110C22×C8—42no—
111C22×C8—42no—
112C22×C8—42no—
113C4×Q8—41no—
114C4×Q8—41no—
115C4×Q8—41no—
116C4×Q8—41no—
117C4×D4—41no—
118C4×D4—41no—
119C4×D4—41no—
120C4×D4—41no—
121C4×D4—41no—
122C4×D4—41no—
123C4×D4—41no—
124C4×D4—41no—
125C4×D4—41no—
126C4×D4—41no—
127C4×D4—41no—
128C4×D4—41no—
129#117—41no—
130#117—41no—
131#117—41no—
132#117—41no—
133#117—41no—
134#117—41no—
135C2×C4⋊C4C22⋊C411yescharacteristic
136C2×C4⋊C4C22⋊C411yescharacteristic
137C2×C4⋊C4C4⋊C411yescharacteristic
138C2×C4⋊C4C4⋊C411yescharacteristic
139C2×C4⋊C4C2×D411yescharacteristic
140C2×C4⋊C4C2×Q811yescharacteristic
141C2×C22⋊C4C22⋊C411yescharacteristic
142C2×C22⋊C4C22⋊C411yescharacteristic
143C2×C22⋊C4C4⋊C411yescharacteristic
144C2×C22⋊C4C4⋊C411yescharacteristic
145C2×C22⋊C4C22×C411yescharacteristic
146C2×C22⋊C4C22×C411yescharacteristic
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—21no—
159C2×C42C22⋊C411yescharacteristic
160C2×C42C22⋊C411yescharacteristic
161C2×C42C22⋊C411yescharacteristic
162C2×C42C22⋊C411yescharacteristic
163C2×C42C4⋊C411yescharacteristic
164C2×C42C4⋊C411yescharacteristic
165C2×C42C4⋊C411yescharacteristic
166C2×C42C4⋊C411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C4⋊C8—42no—
170C4⋊C8—42no—
171C4⋊C8—42no—
172C4⋊C8—42no—
173C4⋊C8—42no—
174C4⋊C8—42no—
175C4⋊C8—41no—
176C4⋊C8—41no—
177C4⋊C8—41no—
178C4⋊C8—41no—
179C22⋊C8—24no—
180C22⋊C8—24no—
181C22⋊C8—44no—
182C22⋊C8—42no—
183C22⋊C8—42no—
184C22⋊C8—44no—
185#97—42no—
186C4×C8—42no—
187C4×C8—42no—
188C4×C8—42no—
189C24#9511yescharacteristic
190C24#9511yescharacteristic
191C24C2×C4211yescharacteristic
192#41—44no—
193#41—44no—
194#41—44no—
195#41—44no—
196C2×Q8—22no—
197C2×Q8—22no—
198C2×Q8—41no—
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—22no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—41no—
206C2×D4—41no—
207C2×D4—41no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—81no—
238C22×C4—81no—
239C22×C4—81no—
240C22×C4#9511yescharacteristic
241C22×C4#9511yescharacteristic
242C22×C4#9511yescharacteristic
243C22×C4#9511yescharacteristic
244C22×C4#9511yescharacteristic
245C22×C4#9511yescharacteristic
246C22×C4#10012yes—
247C22×C4#10112yes—
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C4⋊C411yescharacteristic
252C22×C4C2×C4⋊C411yescharacteristic
253C22×C4C2×C4⋊C411yescharacteristic
254C22×C4#13112yes—
255C22×C4#13112yes—
256M16—84no—
257M16—84no—
258C2×C8—44no—
259C2×C8—44no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—84no—
267C2×C8—84no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C42—21no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42—21no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—21no—
289C42—21no—
290C42—21no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42#9911yescharacteristic
299C42#9911yescharacteristic
300C42#9911yescharacteristic
301C42#9911yescharacteristic
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23#34312yes—
316C23C2×#9511yescharacteristic
317C23#40512yes—
318Q8—44no—
319D4—44no—
320D4—44no—
321D4—44no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4#34211yescharacteristic
368C2×C4#34211yescharacteristic
369C2×C4#34211yescharacteristic
370C2×C4#34211yescharacteristic
371C2×C4#34312yes—
372C2×C4#40811yescharacteristic
373C2×C4#40811yescharacteristic
374C2×C4C2×#9911yescharacteristic
375C2×C4C2×#9911yescharacteristic
376C2×C4C2×#10011yescharacteristic
377C2×C4C2×#10111yescharacteristic
378C2×C4#42412yes—
379C8—84no—
380C8—84no—
381C8—82no—
382C8—82no—
383C22—21no—
384C22—21no—
385C22—22no—
386C22—22no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—81no—
399C22—81no—
400C22#172911yescharacteristic
401C22C2×#34211yescharacteristic
402C22C2×#34311yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—22no—
407C4—22no—
408C4—41no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4#139112yes—
416C4#147111yescharacteristic
417C4#147111yescharacteristic
418C4#147211yescharacteristic
419C4#147211yescharacteristic
420C2—21no—
421C2—21no—
422C2—42no—
423C2—42no—
424C2—81no—
425C2#791511yescharacteristic
426C2C2×#139111yescharacteristic
427C2#1257911yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC24C2×C42
Frattini, cosocleC2×#117C23
socleC22C2×#342
prime radicalC22×#41C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #7188C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#206292
#173612
#173492
#71882